Normalizing the hydrogenic polar solutions $$\Theta _{\ell m}(\theta )$$ without Associated Legendre polynomials
IF 1.7
3区 化学
Q3 CHEMISTRY, MULTIDISCIPLINARY
引用次数: 0
Abstract
The normalization of the polar functions \(\Theta _{\ell , m} (\theta )\) for the solution of Schrödinger’s equation for the Coulomb potential usually proceeds by appealing to the properties of Associated Legendre polynomials. We show how to achieve the normalization directly from the overall form of the solution and the recursion relation for its series part. When combined with a previous such normalization for the radial part of the solution, the entire hydrogen atom solution can be normalized without having to invoke any properties of special functions.
对氢极解进行归一化处理 $$\heta _{ell m}(\theta )$$ 无需关联 Legendre 多项式
对于库仑势的薛定谔方程的解，极值函数 \(\Theta _{\ell , m} (\theta )\) 的归一化通常是通过求助于相关的 Legendre 多项式的性质来实现的。我们展示了如何直接根据解的整体形式及其序列部分的递推关系实现归一化。结合之前对解法径向部分的归一化，整个氢原子解法就可以归一化，而无需调用任何特殊函数的性质。
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来源期刊
CiteScore
3.70
自引率
17.60%
发文量
105
审稿时长
6 months
期刊介绍：
The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches.
Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.
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